Fundamental algorithms for cryptography

Elliptical curves and isogenies

  • Effective arithmetic of finite bodies and their extensions

– Extension towers, composita, morphisms.
– Algebraic closure of GF (p).
– Geometric applications: torsion, isogenies, Tate module.


  • Isogenies, couplings

– Calculation of explicit isogenies: Vélu formulas, Elkies formulas, higher genus.
– Isogeny volcanoes, isogeny graphs.


  • Cryptographic applications

– Homomorphic encryption.
– Post-quantum cryptography.
– Cryptanalysis.

Advanced network reduction algorithms

  • Since the invention of LLL, many improvements have been proposed:

– Algorithms which provide reduced databases of the same quality as LLL but faster -> study their implementation on massively parallel computers.

– Algorithms which aim to produce better quality networks than LLL -> obtain bases whose first vector is the shortest vector of the network.


  • Implementation of network reduction attacks

– Build an effective and highly configurable toolbox integrating all the standard cryptanalyses using network reduction, including those based on finding small roots using the Coppersmith method.